Lane–Emden problems: Asymptotic behavior of low energy nodal solutions

Grossi, Massimo; Grumiau, Christopher; Pacella, Filomena

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 1, pp. 121-140.

We study the nodal solutions of the Lane–Emden–Dirichlet problem

$$\{\begin{array}{cc}-\Delta u={\left|u\right|}^{p-1}u,\hfill & \text{in}\Omega ,\hfill \\ u=0,\hfill & \text{on}\partial \Omega ,\hfill \end{array}$$

where Ω is a smooth bounded domain in ${ℝ}^2$ and $p>1$. We consider solutions $u_p$ satisfying

$$p\underset{\Omega }{\int }{|\nabla u_p|}^2\to 16\pi e\text{as}p\to +\infty$$

and we are interested in the shape and the asymptotic behavior as $p\to +\infty$.First we prove that (⁎) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that $p{u}_p$ can be characterized as the difference of two Greenʼs functions and the nodal line intersects the boundary of Ω, for large p.

Classification : 35J91, 35B32
Mots clés : Superlinear elliptic boundary value problem, Least energy nodal solution, Asymptotic behavior, Variational methods

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